This invention relates to optical metrology, and particularly to the use of interferometry to measure the surface profile of an object with reduced noise sensitivity.
In the use of interferometry to measure the surface profile of an object, the object is illuminated with a reference light beam and the light reflected from the object is caused to interfere with the reference beam so as to produce a two-dimensional interference pattern, or xe2x80x9cinterferogram.xe2x80x9d The interferogram is a function of the two-dimensional distribution of the phase difference between the reference and reflected beams, or xe2x80x9cphase map.xe2x80x9d Since the phase map depends on the optical path difference (xe2x80x9cOPDxe2x80x9d) between those beams, it represents a two-dimensional map of the surface profile of the object.
The interferogram is a two-dimensional distribution of light intensity which varies as follows:                                                                         I                ⁡                                  (                                      x                    ,                    y                                    )                                            =                              xe2x80x83                            ⁢                                                                    w                    r                                    ⁡                                      (                                          x                      ,                      y                                        )                                                  +                                                      w                    t                                    ⁡                                      (                                          x                      ,                      y                                        )                                                                        "RightBracketingBar"                    2                =                              a            2                    +                      b            2                    +                      2            ⁢                                          (                ab                )                                            1                2                                      ⁢            cos            ⁢                          xe2x80x83                        ⁢            2            ⁢                          k              ⁡                              [                                                      s                    ⁡                                          (                                              x                        ,                        y                                            )                                                        -                  l                                ]                                                                            =                  xe2x80x83                ⁢                  1          +                      γcos            ⁡                          [                              φ                ⁡                                  (                                      x                    ,                    y                                    )                                            ]                                          
where wr(x,y)=ae2ikl, the reference wavefront,
wt(x,y)=be2iks(x,y), the wavefront reflected from the object under test,
k=2xcfx80/xcex
xcex is the wavelength of the light,
l is an arbitrary measure of the reference beam path length,
s(x,y) is the surface profile of the object under test,
y=2ab/(a2+b2), the interference fringe visibility, and
xcfx86(x,y) is the phase difference between the reference beam and the beam reflected from the object under test due to variations in the surface profile of the object.
Since the reference wavefront is considered to be flat and l is fixed, variations in the surface profile of the object under test, s(x,y), are proportional to the phase difference between the test and reference beams, xcfx86(x,y), and an arccosine function of variations in the intensity of the interference fringes, I(x,y).
Often the average surface height of the object under test is not parallel to the reference beam wavefront. This introduces a linear component in phase difference, xcfx86(x,y), which is known as tilt. It is often desirable to remove the tilt from phase difference measurements as a step in calculating the surface profile from the phase difference measurements.
Typically, the interferogram is imaged onto a video camera or CCD array, which is used to produce a two-dimensional array of measurement points, or pixels. A corresponding two-dimensional array of phase map pixels, xcfx86(xi,yj), is then produced using one of a variety of known methods. The surface profile is calculated from this array of pixels. A system for performing this process is described, for example, in Wyant et al., U.S. Pat. No. 4,639,139, entitled OPTICAL PROFILER USING IMPROVED PHASE SHIFTING INTERFEROMETRY (xe2x80x9cWyant et al.xe2x80x9d), hereby incorporated by reference in its entirety.
A problem in determining the surface profile from the phase map, xcfx86(xi,yj), is that the cosine function repeats, or wraps around, every 2xcfx80 radians of phase difference, that is, every wavelength, xcex, units of OPD. So, for example, one cannot tell the difference between cos xcfx80/4 and cosine 5xcfx80/4. Since one wavelength of the light used to produce the interferogram is often less than the cumulative variations in object surface height from one pixel to the next, sometimes even less than the change in surface height between one pixel and the next pixel, and often less than the difference in OPD introduced by tilt, it is necessary to account for the discontinuities caused by the aforedescribed wrapping. Doing so is called xe2x80x9cunwrapping.xe2x80x9d
Another problem in determining the surface profile from xcfx86(xi,yj) is that both the discontinuities caused by wrapping and the sampling to produce pixels, as well as other random variations in the phase difference data, produce noise in the phase map, xcfx86(xi,yj), which can produce measurement errors. It is desirable to reduce the sensitivity to that noise.
There are several methods used to calculate a surface profile map from a phase difference map. They differ mainly in the way that they handle noise.
One such method, sometimes referred to as xe2x80x9cstandard integrationxe2x80x9d, scans through the phase data consecutively and removes each half-wave discontinuity as it is encountered by adding or subtracting 2xcfx80 radians of phase to the adjusted previous adjacent pixel to adjust the phase of the current pixel, if needed. This step is adequate as long as there is no noise and the phase difference between adjacent pixels is less than 2xcfx80 radians. However, there usually is noise, and discontinuities greater than 2xcfx80 radians may be introduced by discontinuities in surface height, and regional under sampling of the fringe pattern, that is, where the sample frequency is such that the absolute value of the phase difference between two pixels is 2xcfx80 radians or more. In the case of discontinuities additional steps must be taken, such as those described in D. Malacara, et al., Interferogram Analysis for Optical Testing, pp. 381-407 (1998). After the phase is unwrapped, tilt is then removed.
A problem with standard integration is that noise tends to propagate from one pixel to the next. This is because each phase adjustment is made with respect to the next preceding pixel. Thus, if there is a noise error in a pixel, that error will affect all of the subsequently evaluated pixels.
In standard integration, noise is reduced by relying on the fact that high noise data typically has a low modulation value. By raising the modulation threshold, most of the noise can be eliminated. The modulation value is the term 2(ab)xc2xd in the interferogram intensity pattern equation shown above.
Another method for unwrapping, commonly known as xe2x80x9cenhanced integrationxe2x80x9d, uses the standard integration steps but adds some data processing to reduce noise that gets through despite raising the modulation threshold. Errors in this method normally result in a streak of pixels running in the x or y direction at the wrong height.
In yet another method for unwrapping, commonly known as xe2x80x9croughness filtering,xe2x80x9d the phase data is first separated into xe2x80x9crough,xe2x80x9d xe2x80x9csmooth,xe2x80x9d and xe2x80x9ccliffxe2x80x9d categories, then treated according to its category. This categorization is accomplished by comparing height differences between adjacent pixels to empirically obtained standards. Cliff data is that which represents at least a halfwave discontinuity. Smooth data is noise free. Rough data does not fit into either of the other two categories and is ignored.
In the measurement of the surface contour of relatively flat surfaces, such as the surfaces of a magnetic disk recording head, noise is the prevalent problem. Surface height variations do not tend to exceed one-half wavelength, though tilt is typically present, but noise often causes a discontinuity to be introduced during the unwrapping. It has been found that because the aforementioned unwrapping methods adjust the phase based on the next-preceding pixel, they are sensitive to noise and produce errors in measuring the surface contour of such relatively flat objects. Therefore, there is a need for reduced noise sensitivity method and apparatus for converting an interferogram phase map to a surface profile map.
The present invention addresses the aforementioned phase unwrapping problem and meets the aforementioned need for reduced noise sensitivity in the measurement of the contour of objects, particularly relatively-flat objects.
Assuming either that the phase data has no significant tilt or that the phase data has been adjusted to remove phase discontinuities due to tilt, a histogram is created from the phase data wherein the bins of the histogram represent phase values from zero to 2xcfx80 radians relative to a reference beam and the items in the bins represent occurrences of respective phase values in the phase data. Useful phase data is then identified by selecting groups of bins whose contents exceed a threshold value. Where useful data wraps around the end of the histogram to the beginning thereof, a selected amount of phase shift is added to the phase data so as to move all of the useful data within the ends of the histogram and thereby avoid errors from discontinuities. Ordinarily the amount of phase shift added is that fraction of 2xcfx80 radians which corresponds to the bin of the histogram having the least number of occurrences therein.
Where tilt is present, additional processing is first used to remove the tilt. The phase data is first differentiated to produce slope data. A slope histogram of the differentiated phase data is then created wherein the bins of the histogram represent slope values and the items in the bins are occurrences of corresponding slope values in the differentiated phase data. A best-fit amount of tilt is determined from the slope histogram by calculating the average slope between two points on the histogram curve that are a predetermined percentage of the maximum height of the curve. Thereafter, the best-fit tilt is subtracted from the original phase data to eliminate tilt.
Therefore, it is a principal objective of the present invention to provide a novel and improved method and apparatus for unwrapping phase data in measuring the profile of a surface using interferometry.
It is another objective of the present invention to provide a method and apparatus for unwrapping phase data in the measurement of the profile of a relatively flat surface using interferometry.
It is a further objective of the present invention to provide a method and apparatus for unwrapping phase data in measuring the profile of a surface using interferometry wherein sensitivity to noise in the phase data is reduced.